On the cohomologies of the de Rham complex over weighted isotropic and anisotropic Hölder spaces

Abstract

We consider the de Rham complex over scales of weighted isotropic and anisotropic Hölder spaces with prescribed asymptotic behavior at infinity. Starting from theorems on the solvability of the system of operator equations generated by the de Rham differential d and the operator $d^{\ast }$ formally adjoint to it, a description of the cohomology groups of the de Rham complex over these scales was obtained. It was also proved that in the isotropic case the cohomology space is finite-dimensional, and in the anisotropic case the general form of an element from the cohomology space is presented.

COMMUNICATED BY:

• O. Celebi

Keywords:

• Weighted Hölder spaces
• integral representation's method
• anisotropic spaces

AMS Subject Classifications:

• 35F35
• 47A53
• 58A12

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Foundation for the advancement of theoretical physics and mathematics ‘BASIS’.